An educated guess we are able to perform is that low rank personnel and servants in a castle or palace were mixed in gender, although some aspects of it were definetely distinguished bewtween men and women.

As long as the tasks to be performed were menial (carrying water, moving stuff around, hauling luggage, lighting fires, cleaning etc.) anyone close by would have done that as these were not specific in any form or way we are able to undestand. Some chores which were instead very specific and assigned to a specific gender were cooking, laundering and most times weaving.

Cooks in medium to high-status households were all men, a tendency reinforced during the XV and XVI centuries. The reasoning behind it was that the kitchen was a place were only the most trustworthy servants were allowed as it was the place were food and precious ingredients (think spices) were kept and prepared. Poisoning as a means of murder was a rather common occurence, or at least a very alive concern, thus who handled dishes and their preparation had to be completely loyal, excluding women outright according to the Middle Ages' conception of things. To name a couple, Bartolomeo Scappi (1500-1577) was the cook of several popes and authored a very useful illustrated recipe book named Opera de l'Arte del Cucinare ("Work on the Art of Cookery") listing many dishes and images of the ideal setting of a kitchen. Also, the anonymous writer of the Liber de Coquina ("Book of Cookery") plausibly composed at the Angevin court of Naples between 1284 and 1314, where the Latin text uses the term "magister", "master" to describe the person doing the cooking. To this day, in modern Italian the furniture where dishes and ingredients can be placed is called "credenza", most likely stemming from the Latin verb "credo", "to have faith [in someone/something]".

Lower status households would have seen the wife or another female family member doing the cooking and food preparation, as it was more apt for those social conditions. We cannot say that it was an absolute occurrence, however, as some XVII century illustrations and paintings show fathers and elderly men tending to kids and doing other chores. But as peasant work goes, men and women ploughed the fields and harvested crops all the same. Craftmanship had some instances of female workforce being used, most of which I'm aware of focus on the city-states of Central and Northern Italy, where already in the mid-1200s there was a prototypical textile industry employing house production for the most basic tasks of wool processing, where also women would be employed. Adding to this, especially poorer housewives scraped a meagre living by performed those jobs consisting of a "makeshift economy": sub-renting beds or chambers, doing small reparations on clothes, taking care of the neighbour's children and things like that.

Marrying and having children was a big concern and social task for both men and women, but many times women would look after other people's houses and even other persons' children too. Wet and dry nurses were very common for a whole set of reasons, chief of these, we assume, were the need to shorten the infertile lactation period as much as possible or the inability of the mother to properly feed the child (either because of her denutrition or perhaps because her work was not compatible with it). Among these, wealthy women didn't like that their pregnancies ruined their bodies (teeth wounds around the nipples or enlarged breasts not fitting with the era's standards of beauty are to be taken into account). Many religious and lay institutions provided both wet and dry nurses for the poor and the orphans as well, and we can tell that these nurses were paid a rather decent salary for their work in most cases.

Laundry and weaving were almost all-female occupations, creating lasting images of women performing these two actions. As far as weaving goes, producing the cloth required to make clothing, blankets, decorative elements, underwear etc. is connected to the need to prepare both an everyday commodity and assembling the elements of the dowry and eventual daughter would need (in Italy, I've found it consists mostly of bed sheets, clothing, napkings, tablecloths, socks etc.). It also symbolized the honest woman who had a house, a husband and children, almost counterbalanced by the prostitute whom instead didn't weave as she didn't have any of the aforementioned elements. This, alongside entries found in registries, has sparked the hypothesis that many of the welfare institutions mentioned above might have provided the nubile and orphaned girls the tools and skill to be able to weave.

This answer focuses mostly on the XIII-XVI centuries, as they are my specialty. I hope it helps you.